ON THE CONDUCTION OF HEAT. 



11 



more rapidly than it would if Newton's law were true. Nothing else is 

 altered. (Theory of Heat, p. 75.) 



This result, it must be confessed, is deduced from the omission of many 

 terms in the equation in order to effect an approximation. It can hardly be 

 regarded as a tolerable expression of fact. 



III. M. Poissons hypothesis. That extra-radiation follows Dulong and 

 Petit's law, whilst conduction follows an analogous law, — the flow of tempe- 

 rature depending on the product of the variation of temperature, and a 

 function of the temperature. In M. Poisson's large work will be found the 

 solution of a considerable number of the resulting equations ; but the solu- 

 tion is in general approximate, and effected in such a manner as to reduce 

 the hypothesis actually to M. Fourier's hypothesis. We find very few prac- 

 ticable results in the work, derived from the proper axioms on which it pro- 

 fesses to be founded. One only can be given which will serve the purpose 

 we have in view. The equation which gives the permanent state of heat in 



an 



uniform prismatic bar is — ( ^ -j — I = — p (y -- '(); (Art. 1 1 8.) 



d V 

 where k and p are functions of the temperature, such that k j— and p v 



respectively represent the flow of heat in a small time (considered as unity) 

 within and on the surface of the body ; <^ is the temperature of the surround- 

 ing vacuum. 



M. Poisson assumes that u — i^ or v is small, or perhaps rather (as appears 

 to us) that certain multipliers of this quantity are small, so that k and p 

 may be expanded in terms of v, and high powers of this quantity may be 

 omitted. He thence deduces the following equation as expressing the per- 

 manent temperature of a bar heated at one extremity ; the length of the bar 

 being indefinitely great : 



8. »= 1 (y -2 m) 0e~^*' H — - (y - 2 m) e ~^^ "^ ; where y 



= log^ 1-0077, and m is undetermined, but depends on the manner in which 

 the conductivity varies with the temperature. (Theorie de la Chaleur, Art. 

 105.) 



On this result we must make some remarks. In the first place, we con- 

 ceive that by a slip of the pen M. Poisson has given a wrong value to y ; 



it appears to be properly -— log 1*0077. The quantities y and m result 



2 e 

 from the expansion ofp and k in the form p + p y u, k + k m u. In the next 

 place, we find some difficulty in understanding what the quantity m actually is 

 supposed to be. If M. Poisson conceives, as he leads us to believe in his pre- 

 face (p~. 6), that Dulong and Petit's law is applicable to the interior as well 

 as to the exterior flow of heat, we cannot see how he regards m as undeter- 

 mined. We should have thought, that if it be admissible to expand in terms 

 of V at all, m must have been known. Perhaps M. Poisson saw a difficulty 

 in admitting this, arising from the circumstance that y — 2 m might (and it 

 appears to us it would) turn out to be negative, and thus disprove the whole 

 theory. The phrase which M. Poisson makes use of is this : Quant a la con- 

 stante m, elle depend de la maniere dont la conducibilite varie avec la tempe- 

 rature ; et sa valeur n' est pas non plus connue. (p. 255.) 



IV. The theory suggested by the author of this report adopts all the re- 

 sults of the hypothesis of Fourier. In this case v will not express the fem- 

 perature as measured by an air-thermometer, but a certain function of that 

 temperature. We have already given the function which appears to us to be 

 the proper one. It is deduced by the following reasoning. Let represent the 



